Multiple choice

Find the number of ways in which $12$ different flowers can be arranged in a garland so that $4$ particular flowers are never separate.

  1. 483886

  2. 483833

  3. 483840

  4. 483777
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C Correct answer
Explanation

Arranging 12 flowers in a garland: (n-1)! / 2. Treating 4 flowers as one unit, we have 9 units (8 + 1). Arrangements = (9-1)! / 2 = 8! / 2 = 40320 / 2 = 20160. Within the unit, the 4 flowers can be arranged in 4! = 24 ways. Total = 20160 * 24 = 483840.

AI explanation

To keep the 4 particular flowers together in a circular garland, we first group them as one unit, leaving 8 other flowers for a total of 9 units. The 9 units can be arranged in a garland in (9 minus 1) divided by 2 factorial ways, which is 20160 ways. The 4 particular flowers can be arranged among themselves in 4 factorial ways, or 24 ways. Multiplying these together gives 20160 multiplied by 24, which equals 483840.

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